Time dependence in $B\to V\ell\ell$ decays

We discuss the theory and phenomenology of $B_{d,s}\to V(\to M_1M_2)\ell\ell$ decays in the presence of neutral-meson mixing. We derive expressions for the time-dependent angular distributions for decays into CP eigenstates, and identify the relevant observables that can be extracted from time-integrated and time-dependent analyses with or without tagging, with a focus on the difference between measurements at $B$-factories and hadronic machines. We construct two observables of interest, which we call $Q_8^-$ and $Q_9$, and which are theoretically clean at large recoil. We compute these two observables in the Standard Model, and show that they have good potential for New Physics searches by considering their sensitivity to benchmark New Physics scenarios consistent with current $b\to s\ell\ell$ data. These results apply to decays such as $B_d\to K^*(\to K_S\pi^0)\ell\ell$, $B_s\to \phi (\to K_SK_L)\ell\ell$ and $B_s\to \phi(\to K^+K^-)\ell\ell$.


Introduction
Rare B decays mediated by flavour-changing neutral currents constitute a unique playground to test the Standard Model (SM) and search for New Physics (NP). Among these, processes mediated by the quark-level b → s transition have received a great deal of attention following a large programme of measurements at B-factories, LHCb and CMS: branching ratios, CP asymmetries and angular distributions of B → K ( * ) µ + µ − [1][2][3][4][5][6][7][8][9][10][11] and B s → φµ + µ − [12] decays, the branching ratio B(B s → µ + µ − ) [13][14][15], and inclusive B → X s observables [16,17]. Global fits to all b → sγ and b → s data have recently uncovered a pattern of tensions between theory and experiment, triggered by the the analysis of the B → K * µ + µ − angular distribution [18][19][20][21], and followed by the measurement of the ratio R K ≡ B(B → Kµµ)/B(B → Kee), which is consistent with New Physics in B → K * µµ data and hints at lepton-flavour non-universality [22][23][24][25][26][27][28]. In this context, the study of new independent b → s observables is of great interest, as a means to gather evidence for (or against) these tensions, and to fingerprint the resulting New Physics.
When dealing with decays of neutral B mesons, experimental observables are affected by particle-antiparticle mixing (oscillations), with the decaying meson being either a B or aB depending on the time of decay. In the case of flavour-non-specific decays -such as decays into CP eigenstates-in which the final state can arise from the decay of both B andB mesons, the mixing and decay processes interfere quantum-mechanically, leading to interesting phenomenological consequences (for a review see for instance Refs. [29,34]). In particular, new observables arise compared to the case without mixing. These observables depend on the experimental set-up (B-factory or hadronic machine), the presence of flavour tagging of the decaying B-meson, and the possibility to perform time-dependent measurements (in contrast to the limitation to time-integrated observables). In the case of b → s transitions, these effects have been so far taken into account in the untagged time-integrated measurements of B s → φµ + µ − [30] and B s → µ + µ − [31] at the LHC, (see also the discussion in Ref. [32] in the case of B s → V V decays). Time-dependent angular analyses of B d,s → V with tagging are much more challenging experimentally, but might be reached at a high-luminosity flavour factory such as Belle-II [33].
In this paper we develop the theoretical framework and study the phenomenological advantages of time-dependent B d,s → V decays, spelling out the new observables that can be accessed, as well as the opportunities for New Physics searches, both at B-factories and hadronic machines. While the formalism is valid for any decay of the type B d,s → V (→ M 1 M 2 ) with M 1 M 2 a CP eigenstate, we identify the following modes of interest: As a summary of the main points to be discussed below, we shall see that: • In the presence of mixing, the time-dependent angular distributions exhibit a new type of angular coefficients, h i and s i , apart from the usual coefficients accessible from flavour-specific decays, J i andJ i , c.f. Eqs. (25), (26).
• Time-integrated CP-averaged rates and CP-asymmetries, as measured at hadronic machines, are affected by mixing effects in two ways, c.f. Eqs. (33), (34): 1) The terms with J i ±J i are multiplied by the factors 1/(1−y 2 ) and 1/(1+x 2 ) respectively, with y = ∆Γ/(2Γ) and x = ∆m/Γ. 2) New contributions proportional to the coefficients h i and s i arise. At B-factories only the first type of corrections appear, and time-integrated quantities are independent of the coefficients s i and h i , c.f. Eqs. (35), (36).
• We identify s 8 and s 9 as new observables of interest, which can be extracted most conveniently from a time-dependent analysis with flavour tagging. Theoretically clean observables can be built from s 8 and s 9 ; two such observables are Q − 8 and Q 9 , which are clean at large hadronic recoil. These observables contain independent information, not accessible from flavour-specific decays. Such observables could be studied, for instance, at a high-luminosity flavour factory, where a separation between B andB samples would be possible together with a study of time dependence of the decay process.
• The observables Q − 8 and Q 9 can be predicted in the Standard Model with small uncertainties (see Fig. 1). In particular, Q 9 measures right-handed currents, with Q SM 9 − cos φ to a very good precision, with φ the mixing angle of the B q system. In addition, these observables are very sensitive to New Physics scenarios consistent with current b → sγ and b → s data, such as models with Z bosons with vector and/or axial couplings to fermions.
The structure of this article is the following. We begin in Section 2 with a discussion on time-dependent angular distributions: In Section 2.1 we review the basic facts of B → V decays without mixing. In Section 2.2 we address the CP parities associated to transversity amplitudes for B → V decays into CP eigenstates. In Section 2.3 we derive the expressions for the time-dependent angular distributions, and identify the new angular observables h i (s) and s i (s) that arise in the presence of mixing, demonstrating in Section 2.4 that s 7,8,9 (s) contain independent information not accessible from the angular distribution of flavour-specific decays. In Section 3 we discuss in detail the two types of observables that can be obtained from the distributions in the presence of mixing: timeintegrated (Section 3.1) and time-dependent (Section 3.2) observables. We also define the observables Q − 8 and Q 9 , which are form-factor-independent at large recoil, and we provide simplified expressions at the leading order of the effective theory in this limit. Standard Model predictions for these observables and New Physics opportunities are discussed in Section 4. Finally, we conclude in Section 5. Some details are relegated to the appendices. In Appendix A we discuss the kinematics of CP-conjugated B → V (→ M 1 M 2 ) decays in terms of momentum invariants and the different conventions for the kinematic angles that appear in the angular distributions. In Appendix B we recall the determination of the CP parity for the different transversity amplitudes. In Appendix C we collect the expressions for the coefficients h i (s) and s i (s) in terms of transversity amplitudes. We first recall a few elements of the analysis of the exclusive b → s decays of the type B → V (→ M 1 M 2 ) . In this subsection, we consider a situation where no mixing occurs, and where the M 1 M 2 state is not (necessarily) a CP-eigenstate. This process is described by the usual effective Hamiltonian, with SM operators plus (potentially) NP operators with chirality flip, scalar or tensor structure [39,42]: where λ q = V qb V * qs and I = {3, 4, 5, 6, 8, 7, 7 , 9, 9 , 10, 10 , S, S , P, P , T, T }. The operators O 1,.., 6 and O 8 are hadronic operators of the type (sΓb)(qΓ q) and (sγ µν T a P R b)G a µν respectively [43], and contribute to b → s processes through a loop coupled to an electromagnetic current (via b → sγ * → s ). These operators are not likely to receive significant contributions from NP, as these would show up in non-leptonic B decay amplitudes 1 . The operators O 7 ( ) ,9 ( ) ,10 ( ) ,S ( ) ,P ( ) ,T ( ) are given by: In the SM, and at a scale µ b = O(m b ), the only non-negligible Wilson coefficients concerning the operators of Eq. (2) are C SM 7 (µ b ) −0.3, C SM 9 (µ b ) 4 and C SM 10 (µ b ) −4 (see Table 2); but all might be affected by NP. Contributions to B → V from electromagnetic dipole operators O 7 ( ) are (like hadronic contributions) of the type b → sγ * → s . Contributions from semileptonic operators O 9 ( ) ,10 ( ) ,S ( ) ,P ( ) ,T ( ) are factorizable and their matrix elements can be written as where M denotes a collection of Lorentz indices. It is clear that all hadronic, dipole, and semileptonic contributions can be recast as decays of the form where N has the quantum numbers of a boson, whose coupling pattern is determined by the operators arising in the effective Hamiltonian. In the SM, the structure of O 7 , O 9 , O 10 shows that N are spin-1 particles, coupling to both left-and right-handed fermions. This is in agreement with the presence of γ * and Z penguin contributions, but it is also able to reproduce the contribution from box diagrams involving two W bosons and a neutrino ((V − A)(V − A) structure in the SM). In an extension of the SM yielding scalar (tensor) operators, one should add N bosons with spin 0 (spin 2 respectively). We will work under the following assumptions, inspired by the situation in the SM and in its most usual extensions • CP might be violated in the decay B → V N , but it is conserved in the decay N → + − .
• N can have spin 0 or spin 1, but not spin 2 (no tensor currents in the effective Hamiltonian).
It proves useful to analyse such decays in terms of transversity amplitudes. Let us call the helicity amplitudes for this decay, where m and n denote the polarisations of the meson V and the virtual boson N decaying into the dilepton pair, respectively. If N has spin 1, as the initial decaying particle has spin 0, the only combination of helicity amplitudes allowed are (m, n) = (0, 0), (+1, +1), (−1, −1), (0, t), where t denotes the timelike polarisation. One can then define the transversity amplitudes [39,45,49] The spin-1 N particle couples to the lepton pair either through¯ γ µ P L or¯ γ µ P R , and we can further separate left-from right-handed components in the amplitudes: On the other hand, due to current conservation and the structure of the time-like polarisation * ν N (t) ∝ (p + +p − ) ν , one can see that A t corresponds to a pure axial coupling to the lepton pair, vanishing in the massless limit. In the case where N is spin 0, the only combination of helicity amplitudes allowed is (m, n) = (0, 0). The effect of a spin-0 particle with a pseudoscalar coupling to leptons can be absorbed into A t , whereas a scalar coupling requires a new amplitude, called A S .
The spin-summed differential decay distribution is given by [45,65] J 1s sin 2 θ K + J 1c cos 2 θ K + J 2s sin 2 θ K cos 2θ l +J 2c cos 2 θ K cos 2θ l + J 3 sin 2 θ K sin 2 θ l cos 2φ + J 4 sin 2θ K sin 2θ l cos φ +J 5 sin 2θ K sin θ l cos φ + J 6s sin 2 θ K cos θ l + J 6c cos 2 θ K cos θ l +J 7 sin 2θ K sin θ l sin φ + J 8 sin 2θ K sin 2θ l sin φ + J 9 sin 2 θ K sin 2 θ l sin 2φ , in terms of the invariant mass of the lepton pair s, and three kinematical angles θ , θ K , φ (see Appendix A). The coefficients of the distribution J i (s) contain interferences of the form Re[A X A * Y ] and Im[A X A * Y ] between the eight transversity amplitudes: and are given by where β = 1 − 4m 2 /s. Similar expressions hold for the CP-conjugate decayB →V (→ M 1M2 ) + − , with angular coefficientsJ i involving amplitudes denoted byĀ X , and obtained from the A X by conjugating all weak phases 2 . The form of the angular distribution for the CP-conjugated decay, however, depends on the way the kinematical variables are defined. In the case in which the same conventions are used irrespective of whether the decaying meson is a B or aB, we have (see Appendix A): where f i (θ , θ K , φ) are defined by Eq. (7), and ζ i = 1 for i = 1s, 1c, 2s, 2c, 3, 4, 7 ; ζ i = −1 for i = 5, 6s, 6c, 8, 9 .
We stress that this result arises just from the identification of kinematics of CP-conjugate decays, and does not rely on any intrinsic CP-parity of the initial or final states involved.

CP-parity of final states and decays into CP eigenstates
The separation into transversity amplitudes not only simplifies the analysis of the interference pattern, but also provides amplitudes with final states possessing definite CPparities 3 . In order to determine the CP-parities associated to the different transversity amplitudes we follow the analysis of Ref. [35], where decays of the type B → M N , with M ,N unstable particles, are considered. The details of how to apply the results of Ref. [35] to the B → V decays of interest are provided in Appendix B; here we briefly summarize the main results. We consider the decaysB → M 1 M 2 + − andB →M 1M2 + − , such that M 1 , M 2 are either CP-eigenstates or CP-conjugates, and define the transversity amplitudes: where X = L0, R0, L , R , L⊥, R⊥, t, S. These two sets of amplitudes are related by where η X are the CP-parities associated to the different transversity amplitudes. We find that (see Appendix B) where η = 1 if M 1 , M 2 are CP conjugates (e.g. are CP eigenstates (e.g. K S K L ). Here η(M ) denotes the intrinsic CP-parity of meson M . For the three processes of interest mentioned in the introduction, the combination of intrinsic CP-parities leads always to η = 1. At this point we can classify the angular observables J i whether they combine amplitudes with identical or opposite CP-parities, and whether they involve real or imaginary parts of interference terms: • Real part with identical CP-parities: i = 1s, 1c, 2s, 2c, 3, 4.
• Imaginary part with opposite CP-parities: i = 8, 9. We note that the numbers ζ i defined in Eq. (12) in a different context (identification of the kinematics between CP-conjugate decays) corresponds to the product of the CPparities of the amplitudes involved in the interference term J i .
We now turn to the case of decays into CP eigenstates: B → f CP . In this context, it is useful to define two different angular coefficients J i ,J i which are CP conjugates of J i : • the angular coefficients J i formed by replacing A X by A X ≡ A X (B → f CP ) (without CP-conjugation applied on f CP ), which appear naturally in the study of time evolution due to mixing, where both B andB decay into the same final state f CP .
• the angular coefficientsJ i , obtained by consideringĀ X ≡ A X (B → f CP ) (with CP-conjugation applied to f CP ), which can be obtained from A X by changing the sign of all weak phases, and arise naturally when discussing CP violation from the theoretical point of view.
From the discussion above we have A X = η XĀX , with η X given in Eq. (15). Plugging these amplitudes into the coefficients in Eq. (9), we see that the two types of angular coefficients are related through with ζ i given in Eq. (12). In addition, in the limit of CP conservation, J i =J i . Since the final state is not self-tagging, an untagged measurement of the differential decay rate (e.g. at LHCb, where the asymmetry production is tiny) yields essentially the CP-average whereas the difference between the two decay rates (which can be measured only through flavour-tagging) involves We see that the convention chosen in Eqs. (10),(11) for flavour-tagging modes allows one to treat on the same footing these modes and the modes with final CP-eigenstates, since the same combinations of angular coefficients occur in both cases when one considers the CP-average or the CP-asymmetry in the decay rate. Let us add that this results from a conventional identification between CP-conjugate decays in the case without mixing. This freedom is not present in the presence of mixing where both decays result in the same final state, which must always be described with the "same" kinematic convention, in the sense of a convention that depends only on the final state, without referring to the flavour of the decaying B meson (see Appendix A).

Angular distributions in the presence of mixing
In the case of B decays into CP-eigenstates, where the final state can be produced both by the decay of B orB mesons, the mixing and decay processes interfere, inducing a further time dependence in physical amplitudes (see e.g. Ref. [29,34]). These time-dependent amplitudes are given by, where the absence of the t argument denotes the amplitudes at t = 0, i.e. in the absence of mixing, and we have introduced the usual time-evolution functions with ∆m = M H − M L and ∆Γ = Γ L − Γ H (see Ref. [29]). The values of the different mixing parameters for the three decays of interest are collected in Table 1.
In the presence of mixing, the coefficients of the angular distribution also become time-dependent, as they depend on the time-dependent amplitudes in Eqs. (19), (20). This evolution can be simplified by noting that CP-violation in B q −B q mixing is negligible for all practical purposes 4 , and we will assume |q/p| = 1, introducing the mixing angle φ: This mixing angle is large in the case of the B d system but tiny for B s , see Table 1.
The time-dependent angular coefficients are obtained by replacing time-independent amplitudes with time-dependent ones in Eqs. (9): We consider the combinations J i (t) ± J i (t) appearing in the sum and difference of timedependent decay rates in Eqs. (17), (18). From Eqs. (19), (20) and (24), we get where x ≡ ∆m/Γ, y ≡ ∆Γ/(2Γ), and we have defined a new set of angular coefficients s i , h i related to the time-dependent angular distribution. The coefficients J i , J i can already be determined from flavour-specific decays. The explicit expressions for s i and h i in terms of transversity amplitudes are collected in Appendix C. Time-dependent angular distributions therefore contain potentially new information encoded in the new angular observables s i and h i . These pieces of information will be analysed in the rest of the paper. For the moment a few comments are in order: • The coefficients h i are very difficult to extract, since they are associated with sinh(yΓt) with y very small. In particular, the time dependence of the untagged distribution (17) provides essentially no new information.
• The coefficients s i for i = 1, 2s, 2c, 3, 4, 5, 6s, 6c are given by the imaginary part of amplitude interferences, s i ∼ Im(e iφĀ X A * Y ), and vanish in the absence of complex phases. This is approximately true for B s → V decays in the SM in regions where strong phases are small, e.g. in the region s 1−6 GeV 2 , and if the NP contribution has the same weak phase as the SM. The corresponding coefficients J i +J i do not vanish.
The coefficients s 8 and s 9 have the following expressions (see Appendix C): We have checked by direct calculation that indeed the coefficients s i with i = 8, 9 are tiny in the SM, and that they do not get significant enhancement from NP contributions if new sources of CP violation are not large. We emphasise that the measurement of the coefficients s 8,9 is challenging from the experimental point of view, since the study of J i (t) − J i (t) requires 1) flavour tagging of the original sample to separate B andB at t = 0, 2) the use of appropriate foldings to extract the corresponding angular contributions, identical to the ones used to extract J 8 and J 9 [2,3], and 3) a time-dependent analysis to isolate the sin(xΓt) coefficients.

Symmetries of the distribution
Having identified a few new observables accessible from the time-dependent angular distributions, it remains to be seen if they are truly independent from the observables that can be extracted from angular distributions of flavour-specific decays. The information that can be obtained from the angular distributions depends on the number of independent combinations of interference terms A X A * Y in the angular coefficients. A systematic formalism to determine which combinations can be accessed from the angular distributions alone is the "symmetry formalism" developed in Refs. [40,47]. 5 In the approximation of massless leptons, and neglecting scalar and tensor operators, the angular distributions of flavour-specific decays contain a unitary symmetry, given by the transformation [40]: with U an arbitrary unitary 2 × 2 matrix, and {σ 0 , σ , σ ⊥ } ≡ {1, 1, −1}. Under this group of transformations, J i → J i . This means that from flavour-specific decays, only those combinations of terms A X A * Y that remain invariant under this transformation can be accessed. This approach is useful to eliminate redundancies among observables built from the angular coefficients J i [40,47].
We now identify the transformation properties of the coefficients s i -neglecting weak phases for simplicity. We note that, under the unitary transformation: where From the explicit expressions given in Appendix C, we see that the only coefficients s i that do not remain invariant are therefore s 7 , s 8 and s 9 . This demonstrates that the coefficients s 7 , s 8 and s 9 contain additional information not contained in the usual angular distributions of flavour-specific decays such as

Observables
The expressions in Eqs. (25), (26) for the coefficients of the time-dependent distributions, show that additional structures arise in the presence of neutral-meson mixing. In this context, two different quantities might be considered: time-integrated observables, or observables related to the time dependence. In this section we discuss the two possibilities.

Time-integrated observables
As discussed in Refs. [32,34], time integration should be performed differently in the context of hadronic machines and B-factories. The time-dependent expressions in Eqs. (25) and (26) are written in the case of tagging at a hadronic machine, assuming that the two b-quarks have been produced incoherently, with t ∈ [0, ∞). In the case of a coherent BB pair produced at a B-factory, one must replace exp(−Γt) by exp(−Γ|t|) and integrate over t ∈ (−∞, ∞) [34]. Interestingly, the integrated versions of CP-violating interference terms are different in both settings, and the measurement at hadronic machines involves an additional term compared to the B-factory case: Making contact with experimental measurements requires to consider the total timeintegrated decay rate. The time-dependent rate is given by which after time-integration becomes where I is the usual normalisation considered in analyses of the angular coefficients. The normalised time-integrated angular coefficients at hadronic machines or B-factories are therefore: We see that the interpretation of the time-integrated measurements Σ i from dΓ(B → f CP ) + dΓ(B → f CP ) is straightforward in terms of the angular coefficients at t = 0. Even in the B s case, the smallness of y means that h i will have only a very limited impact on the discussion. The time-integrated terms ∆ i from dΓ(B → f CP )−dΓ(B → f CP ) are subject to two different effects, in particular for B s where x is large: (a) they receive contributions proportional to x and y with a different combination of interference terms (in the case of a measurement at a hadronic machine), (b) they are suppressed (in all experimental set-ups) by a factor (1 − y 2 )/(1 + x 2 ).
The discussion above applies in particular to the measurement of B s → φ(→ K + K − ) as performed at LHCb [12]. Since this is not a self-tagging mode, and assuming that there is an equal production of B s andB s , what is measured is dΓ(B s → φ(→ K + K − ) ) + dΓ(B s → φ(→ K + K − ) ), so these measurements have access to the following combinations: J i +J i Hadronic for j = 1s, 1c, 2s, 2c, 3, 4, 7, J i −J i Hadronic for j = 5, 6s, 6c, 8, 9. (45) The time-integrated observables Σ 6 Hadronic and Σ 9 Hadronic have already been measured (under the name of A 6 and A 9 ) in Ref. [12], and are indeed measuring CP-violation. In the context of the extraction of s 8 and s 9 at hadronic machines, one expects them to dominate ∆ 8,9 Hadronic , especially in the case of B s decays where x enhances their contribution with respect to the (J i − J i ) term. However, they are overall suppressed by a factor ∼ 1/x, which in the case of B s decays is quite effective (1/x ∼ 0.04). In addition, the necessity to perform initial flavour tagging makes these measurements very difficult at hadronic machines. We see therefore that Σ i contain essentially the same information as (J i + J i ), whereas ∆ i have a potentially richer interpretation, but are suppressed and thus probably difficult to extract experimentally. In the following section we will see that time-dependent observables do lead to more interesting opportunities.  (26). We have also seen that the coefficients h i are the sinh(yΓt) coefficient of J i (t) + J i (t), whose effect remains negligible for t 10 τ Bq , constituting a rather difficult measurement.

Time-dependent "optimised" observables with tagging
From the theoretical point of view, these observables are quadratic in hadronic form factors (see Section 4.1). For instance, and similarly for J i (s),J i (s), h i (s). Here F X,Y represent (schematically) hadronic B → V form factors related to the amplitudes A X,Y . These form factors constitute a major source of uncertainty in the theoretical predictions for the observables. This problem is usually tamed by defining a class of special observables with reduced sensitivity to form-factor uncertainties [40,[49][50][51][52][53][54]. These "optimised" observables can be constructed systematically, both in the region of large recoil of the vector meson (s m 2 B ) [40] and at low recoil (s ∼ m 2 B ) [50,54], where the use of effective field theories (SCET [59,60] and HQET [66,67] respectively) ensures a complete cancellation of form factors at the leading order in the respective expansions 6 .
In the following, we focus on the large-recoil region for definiteness. We consider the following optimised versions of the observables s 8, 9 : There are other possible normalizations for s 8 that are also optimised at large recoil: The observable Q + 8 has the particularity of being also optimised at low recoil. However, we find that both Q + 8 and Q 0 8 are slightly less sensitive to NP than Q − 8 . While it might be worthwhile to study these observables further, we will focus here on Q − 8 for illustration, noting that its properties do not differ much from those of Q + 8 , Q 0 8 . Concerning Q 9 , other possible normalizations involve J 6s or J 9 , both of which lead to observables that are optimised also at low recoil. We do not consider this possibilities further as the denominators contain zeroes within the kinematical region of interest.
It is useful to consider these observables in the large-recoil limit (see e.g. Refs. [40,41]), where the expressions simplify considerably, the cancellation of form factors is exact, and the dependence on the Wilson coefficients is apparent. We find: where we have assumed real C i , and used the notation C ± i = C i ± C i . We note that if C i = 0 (that is, C + i = C − i ), on has Q 9 = − cos φ, so that the value of (Q 9 + cos φ) is a measurement of right-handed currents.
In the following section we give Standard Model predictions for these observables and study briefly their sensitivity to New Physics. approximation valid in the region s m 2 B : leading order in α s and leading power in the SCET expansion. This is, of course, a slight abuse of language.  4 Numerical Analysis

Standard Model
The systematic formalism to B → V decays at low hadronic recoil to NLO in QCDfactorisation has been presented in Ref. [55] and is by now quite standard. In our analysis we follow closely the procedure of Refs. [54,58] to which we refer the reader for further details. The different transversity amplitudes can be written as: where M = m Bq and m = m V , and: • The normalizations are given by • Y t (s) and Y u (s) are the 1-loop contributions from 4-quark operators to the photon penguin with the structuresγ µ b, sometimes combined with C 9 into C 9eff (s). Y t (s) denotes the contribution proportional to V tb V * ts , and can be found in Eq.(10) of Ref. [55]. Y u (s) denotes the CKM-suppressed contribution, which is multiplied by the prefactor λ ut (B) = V ub V * us /V tb V * ts or λ ut (B) = V * ub V us /V * tb V ts . This function can be found in Eq. (A.3) of Ref. [56].
• The functions T i encode contributions from dipole operators C 7 ( ) , and the rest of the hadronic contributions not contained in Y t , Y u : The quantities T ⊥, represent factorizable and non-factorizable hadronic contributions in QCD-factorisation and can be extracted from the formulae in Section 2 of Ref. [55]. They depend on distribution amplitudes and on two "soft" form factors ξ ⊥ (s), ξ (s). The "effective" coefficients C ± 7eff include contributions from 4-quark operators with b and s-quark loops with the structures[/ q, γ µ ]b (see e.g. Refs. [44,57]).
• The parameters η entering A L,R 0 . Following Ref. [54], we use the large-recoil symmetry relations [59,60] to express these form factors in terms of ξ ⊥ (s) and ξ (s), defined in the "scheme 1" of Ref. [58], including factorizable power corrections. At a second stage, these are themselves expressed in terms of V (s), A 1 (s), A 2 (s), which are taken from the lightcone sum rule calculation of Ref. [61], both for B → K * and B s → φ transitions.
• The Wilson coefficients C ± i are defined as: Model values for these coefficients are collected in Table 2, computed at a renormalisation scale µ b = 4.8 GeV. In the error analysis we consider a variation of µ ∈ [µ b /2, 2µ b ]. In addition, we have C SM 7 = (m s /m b ) C SM 7 .
Following this procedure, we compute central values and errors (by means of a flat scan over all parameters) for the observables Q − 8 and Q 9 in the SM. We compute these  Figure 1: SM prediction for the observables Q − 8 and Q 9 in the case of B s (t) → φ(→ K + K − )µ + µ − (upper row) and B d (t) → K * (→ K S π)µ + µ − (lower row), in the large-recoil region, including error estimates from all sources. See the text for details. observables differentially in s, keeping in mind that a proper comparison with data would require an integration over bin ranges (see for example the discussion in Ref. [53]).
Our SM results for the observables Q − 8 and Q 9 in the low-s region are shown in Fig. 1. We show both cases: B s (t) → φ(→ KK)µ + µ − and B d (t) → K * (→ K S π 0 )µ + µ − , noting that the results are very similar. We see that indeed the observable Q 9 − cos φ in the whole region, while Q − 8 features a distinctive shape with a zero at s 0 2 GeV 2 . This is located at the same position as the zero of s 8 , which can be expressed solely in terms of Wilson coefficients taking the large-recoil limit, The position of this zero measures a different ratio of Wilson coefficients compared to the zero of other observables, such as A FB or P 2 [40]. We stress that the bands in Fig. 1 include all sources of error including parametric and form-factor uncertainties, as well as our estimates of power corrections, exhibiting the theoretical accuracy for these observables in the Standard Model.

New Physics
We now study the sensitivity of the observables Q − 8 and Q 9 to different models of New Physics. We start with a general scan of (real) New Physics contributions to Wilson coefficients compatible with all current constraints from rare B-decays, in order to asses the NP reach of the new observables. For that purpose, we write The result of this scan is shown in Fig. 2. We consider separately three scenarios: • LHC (Left-Handed Currents) scenario: NP contributions to C 7 , C 9 , C 10 only. This corresponds to the orange regions in Fig. 2, delimited by dashed lines (along the line Q 9 = −1 on the right-hand plot). • RHC (Right-Handed Currents) scenario: NP contributions to C 7 , C 9 , C 10 only. This corresponds to the red regions in Fig. 2, delimited by dotted lines.
• General NP scenario: NP contributions to all six coefficients C 7 ( ) , C 9 ( ) , C 10 ( ) . This corresponds to the regions in green in Fig. 2, with solid borders.
We also show the SM predictions for comparison (blue bands in Fig. 2, with Q SM 9 −1 and Q SM 9 −0.7 for the B s and B d cases respectively). We see that NP can indeed have a large impact on Q − 8 , Q 9 . As discussed in Section 3.2, any significant deviation of Q 9 from Q SM 9 − cos φ requires right-handed currents. We finish our exploratory NP analysis by studying a few motivated benchmark NP scenarios: A. Best fit point in the C 7 − C 9 scenario of Ref. [18]: B. Best fit point in the C 9 − C 9 scenario of Ref. [25]: C NP 9 = −1.28, C NP 9 = 0.47 .
The predictions for the observables Q − 8 and Q 9 within the benchmark scenarios are shown in Fig. 3, together with the SM prediction. We see that, among the considered scenarios, the only ones leading to significant deviations with respect to the SM are scenarios C (corresponding to large NP contributions to C 9 and C 10 ), while scenarios A, B and D are very close to the blue band (corresponding to the SM prediction). As discussed before, scenario C1 has no impact on Q 9 as it has no right-handed currents. Therefore, measurements of these observables compatible with the SM would give support to the best fit points obtained in the global fits of Ref. [18,25] that we have considered, with the potential to exclude the scenarios with C NP 9 ( ) , C NP 10 ( ) such as the one discussed in Ref. [22]. As an alternative viewpoint, these observables could test the latter scenarios, and provide an alternative confirmation if more accurate measurements for time-integrated observables happened to confirm any of them.
In the case where M 1 M 2 is a CP eigenstate (such as , neutral B-meson mixing interferes with the decay, leading to interesting differences with respect to flavour-specific processes where mixing plays no role (such as B d → K * 0 (→ K − π + )µ + µ − ). In this paper we have studied the effects induced by neutral-meson mixing for the analysis of exclusive B → V decays, spelling out the theoretical formalism and analysing its phenomenological consequences.
As a first observation, the angular distributions become time-dependent, with additional structures compared to the case without mixing. These structures are the new angular coefficients h i and s i , defined in Eqs. (25), (26) and given explicitly in terms of the different amplitudes in Appendix C. Two types of observables can then be defined for these modes: time-integrated and time-dependent observables. The first type depend on the experimental set-up (B-factory of hadronic machine) and differ from the corresponding observables in decays without mixing by multiplicative factors depending on the mixing parameters x and y. In addition, the expressions for time-integrated observables at hadronic machines include an extra term proportional to the coefficients h i or s i . This is similar to analogous relations derived for [32] and B s → µ + µ − [31]. The corresponding expressions for time-integrated observables are given in Eqs. (33)- (36). However, it seems difficult to extract h i or s i using time-integrated observables, as they are suppressed by small meson-mixing parameters.
On the other hand, a time-dependent angular analysis with flavour tagging paves the way for the observables s i . We identify s 8 and s 9 as the most interesting observables, as they are expected to be large even in the absence of CP violation. We have demonstrated that these observables contain new information compared to the angular coefficients J i , and we have built "optimised" versions of these observables with reduced sensitivity to form factors. We have focused on two such observables, called Q − 8 and Q 9 and defined in Eqs. (47), (48). These observables can be predicted in the SM with good precision (see Fig. 1), and show good sensitivity to particular New Physics scenarios (see Figs. 2 and 3).
Current analyses of b → s transitions point towards deviations compared to SM expectations, explained via large NP contributions to C 9 (and potentially smaller contributions to other Wilson coefficients). It is particularly interesting and useful to cross check this trend from other sources. Our analysis shows that additional information could come from time-dependent angular analyses of tagged B d,s → V (→ M 1 M 2 ) decays, with M 1 M 2 a CP eigenstate. We thus encourage exploratory studies to determine the experimental feasibility of such analyses, in particular in the context of a high-luminosity flavour factory such as Belle-II.
We see that conventions A and C yield the same relation between dΓ and dΓ, but they apply to different kinds of modes. For decays into flavour-specific modes, convention B is also possible, but with a different relationship between dΓ and dΓ. In the present paper, we choose convention A for decays into flavour-specific modes as well as for B → V (→

B CP-parities associated to transversity amplitudes
We consider the decay B → V N (cf. Eq (4)) where V and N are unstable particles, V decaying into two particles M 1 and M 2 . As shown in Ref. [35], the CP-parity of a final state X is given by with τ = τ (M 1 ) + τ (M 2 ) + τ (N ) the "transversity" of the state M 1 M 2 N (defined below), and ξ depends on the class of decay: • class 1: V (not necessarily with a definite spin) decays into M 1 and M 2 which are CP-eigenstates, and N decays into a CP-eigenstate. In this case, • class 2: V (with a definite spin s V ) decays into spin-0 M 1 and M 2 which are CPconjugates, and N decays into a CP-eigenstate. In this case, • class 3: V and N are CP-conjugates with a definite spin s V , with V decaying into spin-0 M 1 and M 2 . In this case, Here η(V ) and s V are the intrinsic CP-parity and spin of the particle V . The first class is illustrated by the time-dependent analysis of B d → J/ψK * (→ K S π 0 ) [36]. For the class-1 processes (B d → K * (→ K S π 0 ) and B s → φ(→ K S K L ) ) and class-2 process (B s → φ(→ K + K − ) ) of interest, we have where η = −η(M 1 )η(M 2 ) for class-1, and η = 1 for class-2. For all the processes considered here, the combinations of intrinsic CP-parities yield η = 1.
In order to determine the CP-parity of the different transversity states, we have thus to determine η(N ) and τ (N ): • Using the language of Ref. [35], we see that A 0 , A ⊥ , A || , A S,t are respectively associated with the combinations of helicity amplitudes denoted G 1+ 0,0,0 , G 1+ 1,0,0 , G 1− 1,0,0 , G 0+ 0,0,0 , with respective CP parities −ξ, −ξ, ξ, ξ. For our decays, it implies that we should have the following associations (modulo 2): The states corresponding to different transversities can be accessed through the angular analysis of the decay.
• η(N ) can be determined from the assumed quantum numbers of the intermediate boson (spin, polarisation, parity). η(N ) is identical for spin-1 particles with vector or axial couplings, whereas scalar and pseudoscalar N have opposite CP-parities. One can also notice that this constrains the spin of the emitted + − pair. If we denote its angular momentum l and total spin s (either 0 or 1), the P -parity of such a fermion-antifermion pair is given by (−1) l+1 , its C parity by (−1) l+s , so that its CP-parity is (−1) s+1 . Since we assumed that the decay N → + − conserves CPparity, we have η(N ) = (−1) s+1 . We have l = s = 0 corresponding to a pseudoscalar N , l = s = 1 corresponding to a scalar N , l = 0, s = 1 corresponding to a (real) vector/axial N , l = 1, s = 0 corresponding to a time-like vector/axial N (this can be checked from the CP-parity of the corresponding fermion-antifermion currents).
For each amplitude, we can determine the intermediate virtual boson N with the appropriate quantum numbers, the corresponding spin of the lepton pair, the transversity associated, and the CP-parity of the final state, as indicated in Table 3. We see in particular that we agree with the assignments for the class-1 decay B d → J/ψK * (→ K S π 0 ) [36]. In the end, we have η X = η for X = L0, L||, R0, R||, t ; η X = −η for X = L ⊥, R ⊥, S .
We impose τ (N ) = 0 for a vector N with timelike polarisation, to obtain the same CPparity as in the pseudoscalar case. This agrees with the expectation that A t should have the same CP-parity as A 0 .

C Expressions for the coefficients s i and h i
The coefficients s i are given by The coefficients h i are given by In the above expressions, the amplitude A X denotes the amplitude A X (B → f ), without applying CP-conjugation to the final state. One has the relation whereĀ X can be obtained from A X by changing the sign of all weak phases.